Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy

The $q$-sum $x \oplus_q y \equiv x+y+(1-q) xy$ ($x \oplus_1 y=x+y$) and the $q$-product $x\otimes_q y \equiv [x^{1-q} +y^{1-q}-1]^{\frac{1}{1-q}}$ ($x\otimes_1 y=x y$) emerge naturally within nonextensive statistical mechanics. We show here how they lead to two-parameter (namely, $q$ and $q^\prime$) generalizations of the logarithmic and exponential functions (noted respectively $\ln_{q,q^\prime}x$ and $e_{q,q^\prime}^{x}$), as well as of the Boltzmann-Gibbs-Shannon entropy $S_{BGS}\equiv -k \sum_{i=1}^Wp_i \ln p_i$ (noted $S_{q,q^\prime}$). The remarkable properties of the $(q,q^\prime)$-generalized logarithmic function make the entropic form $S_{q,q^\prime} \equiv k \sum_{i=1}^W p_i \ln_{q,q^\prime}(1/p_i)$ to satisfy, for large regions of $(q,q^\prime)$, important properties such as {\it expansibility}, {\it concavity} and {\it Lesche-stability}, but not necessarily {\it composability}.Comment: 9 pages, 4 figure

Resistivity phase diagram of cuprates revisited

The phase diagram of the cuprate superconductors has posed a formidable scientific challenge for more than three decades. This challenge is perhaps best exemplified by the need to understand the normal-state charge transport as the system evolves from Mott insulator to Fermi-liquid metal with doping. Here we report a detailed analysis of the temperature (T) and doping (p) dependence of the planar resistivity of simple-tetragonal HgBa$_2$CuO$_{4+\delta}$ (Hg1201), the single-CuO$_2$-layer cuprate with the highest optimal $T_c$. The data allow us to test a recently proposed phenomenological model for the cuprate phase diagram that combines a universal transport scattering rate with spatially inhomogeneous (de)localization of the Mott-localized hole. We find that the model provides an excellent description of the data. We then extend this analysis to prior transport results for several other cuprates, including the Hall number in the overdoped part of the phase diagram, and find little compound-to-compound variation in (de)localization gap scale. The results point to a robust, universal structural origin of the inherent gap inhomogeneity that is unrelated to doping-related disorder. They are inconsistent with the notion that much of the phase diagram is controlled by a quantum critical point, and instead indicate that the unusual electronic properties exhibited by the cuprates are fundamentally related to strong nonlinearities associated with subtle nanoscale inhomogeneity.Comment: 22 pages, 5 figure

Comparison of light transmission and reflection techniques to determine concentrations in flow tank experiments

Transmissive and reflective intensity measurements for visual concentration determinations in 2D flow tank experiments were compared and evaluated for their applicability in the study of flow and transport phenomena. A density-dependent heterogeneous flow experiment was conducted and transmission and reflection images of the dyed saltwater plume were analyzed. A single light source and dark curtains forced the light to pass through the porous media only, thus facilitating the transmission measurements. The reflection images delivered a more homogeneous spatial illumination than the transmission images. Major perturbations of the transmission images were lens flare effects and light dispersion within the bead-water-Plexiglas system which smear the front of the plume. Based on the conducted evaluation of transmissive and reflective intensity measurements, the reflection data delivered more reliable intensity values to derive solute concentrations in intermediate scale flow tank experiment

Random projections and the optimization of an algorithm for phase retrieval

Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the attractive character of which determines the effectiveness of the algorithm. The behavior of the difference map near fixed points is controlled by the relative orientation of the tangent spaces of the two constraint subspaces employed by the map. Since the dimensionalities involved are always large in practical applications, it is appropriate to use random matrix theory ideas to analyze the average-case convergence at fixed points. Optimal values of the gamma parameters of the difference map are found which differ somewhat from the values previously obtained on the assumption of orthogonal tangent spaces.Comment: 15 page

The Effect of Smart Contracts on Online Investment Decisions: An Experimental Study in ICOs

The imbalance of internal and external knowledge for investments in Initial Coin Offerings (ICO) leads to an information asymmetry, where issuers may further exploit a moral hazard as a resulting mismatch of time and interest during lock-up situations. The existing regulatory vacuum is mirrored by literature, as scholars deliver insights on effective means of signaling. However, research on smart contracts as immutable mechanisms and effective signals to mitigate risks for online investments remains an untapped subject, whilst market demand for solutions to an existing agency problem remains high. To respond to a pressing research question, this study conducted a randomized between-subjects online experiment with a sample of 391 participants. Results include a significant positive effect of the implementation of smart contracts on investor decisions in a present lock-up situation